What do the terms קדוש (Kadosh), קדושים, קדושה, and קדיש really mean?
I understood the term to mean designated, differentiated or set aside for a purpose. So Kiddush-קידוש would be the verbal declaration that designates the day of Shabbat and differentiates it from all the days that precede it (see the Rambam on the Mitzva of Kiddush). The קדושת היום would be the legal designation of the day. The בית המקדש is a place designated to recognize ideas about God. An animal can be made הקדש, thus designating it for the purpose of 'לה (recognizing ideas and עבודת ה). A קדשה is a woman set aside for sexuality (As YS quoted Rashi "הקדשה: מקודשת ומזומנת לזנות"). when we say God is קדוש I believe we mean he is differentiated from all his creations.
The dictionary compiled based on the works of Rav Hirsch gives an extensive list, divided by sense like a modern dictionary. However, the bibliographic element of that entry mirrors closely the listing, divided by binyan, of Ibn Janach's Sefer Hashorashim (Michlol).
קדש - prepare for task, dedicate all resources
2: reserving, prohibiting for use
4: making independent
5: inviting, summoning
6: being extremely moral
7: being immoral
9: home of consecration
Note that items #6 and #7 are not contradictory. Although the usage of #7 in the Torah is probably euphamistic, this is a Biblical dictionary so it is accurate of it to list that as a separate sense of the root.
The very Basic answer is separated from matters of Sexuality and Sin as the first Rashi in Kedoshim says:
קדשים תהיו: הוו פרושים מן העריות ומן העבירה, שכל מקום שאתה מוצא גדר ערוה אתה מוצא קדושה, אשה זונה וחללה וגו' אני ה' מקדשכם, (ויקרא כא ז - ח) ולא יחלל זרעו אני ה' מקדשו (ויקרא כא טו) קדושים יהיו אשה זונה וחללה וגו' (ויקרא כא ו - ז):
The Ramban who first says he does not completely agree with Rashi and says it is a broader thing. similarly says to be separated in things which are allowed to you like wine and he quotes a gemara how the Rabbi would limit themselves in conjugal relations similar to Rashi:
ציווה בדבר כללי שנהיה פרושים מן המותרות. ימעט במשגל, כעניין שאמרו (ברכות כב א): שלא יהיו תלמידי חכמים מצויין אצל נשותיהן כתרנגולין, ולא ישמש אלא כפי הצריך בקיום המצווה ממנו. ויקדש עצמו מן היין במיעוטו, כמו שקרא הכתוב
The one thing they both seem to agree on is it is separation from what is Muttar to you.
The Rambam (Principle 4 of the Rambam’s introduction to the Sefer haMitzvot) says it is a a general statement that describes the effect of doing a mitzvah.
The English translation of קדוש as "Holy" is hopelessly frustrating to me, because it only begs the question. (What is "Holy" supposed to mean?!)
I think Isaiah 6 provides us with the best material to understand the meaning of קדוש. Adonai is exclaimed to be קדוש (triple exclamation) to which Isaiah responds, "Woe to me, I am a man of unclean lips." קדוש therefore has to describe a particular quality of Adonai that evoked Isaiah's recognition of personal uncleanliness and impurity of heart. – My guess, then, is that קדוש means something like "intrinsically pure".
אני צויתי למקודשי תני רב יוסף אלו הפרסיים המקודשין ומזומנין לגיהנם:
"I have commanded to my קדש ones (Yishayahu 13:3)--" Rav Yosef taught: These are the Persians, who are separated (קדש) and prepared foor Gehinnom.
When taken with its usage in Torah, we find that the root, קדש means 'separated for a special cause."